4.5 Discussion
4.5.3 Damped Oval Orbit Model
Here we take another model including non-circular motion due to the Galactic bar to explain the observed radial velocities and the proper motions of our samples and the longitude-velocity diagram of HI. In this subsection, we use a damped oval orbit model with a weak bar potential, basically following Wada (1994), Sakamoto et al.
(1999) and Binney & Tremaine (2008). The damped oval orbit model describes orbits of gas particles in the weak bar potential based on a linearized perturbation. Here, self gravitation of the gas particle is not considered, and an axisymmetric potential and an asymmetric bar potential are supposed. This model approximates the collisional nature of the gas by a damping term, which tend to force the gas motion back to circular rotation.
In the present study, we adopt a axisymmetric potential Φ0 in polar coordinate (R, φ) as follows (Binney & Tremaine (2008)),
Φ0(Rg) = 1
2v02ln(R2c+R2g), (4.9) where Rg is radius of guiding center, Rc is core radius describing a dropping point on a rotation curve, v0 is constant of a speed of the galactic circular rotation. Here we suppose that Rc = 0.01 ≪ 1 and v0 = 220 km s−1 to describe the flat rotation curve with Θ0=220 km s−1. Non-axisymmetric part of a potential Φb (Wada (1994)) as follows,
Φb(Rg) =−ǫ aR2g
(Rg2+ a2)2, (4.10)
where ǫ is a constant value which represents strength of the bar potential, ‘a’ is an arbitrary constant, and supposed a= 1.
A equation of motion for the gas particle in the potential Φ at polar coordinate (R,φ) which rotate with a pattern speed of the bar Ωb is as below,
R¨−Rφ˙2 =−∂Φ
∂R+ 2ΩbRφ˙ + Ω2bR−2γR,˙ (4.11)
Rφ¨+ 2 ˙Rφ˙ =− ∂Φ
R∂φ −2ΩbR,˙ (4.12)
whereγ is damping rate. From these linear simultaneous equations, solutions of the coordinate of the gas particle (R, φ) at each guiding center (Rg,φg) at given time are derived as,
R =Rg+Acos [2(Ω0−Ωb)t+α] =Rg+Acos (2φg+α), (4.13)
φ = (Ω0−Ωb)t− B Rg
sin [2(Ω0−Ωb)t+β] =φg − B Rg
sin (2φg +β), (4.14) wheret is time andφg is (Ω0−Ωb)t. Related equations also listed at the rest for the equation 4.13 and 4.14.
Ω(R)≡ ±√
R−1dΦ0/dR, (4.15)
and,
Ω0 ≡Ω(Rg). (4.16)
Then,
Ω0 = v0
√R2c+R2g, (4.17)
κ0 ≡
√( d2Φ0
dR2 )
Rg
+ 3Ω20 = v0√
4Rc2+ 2Rg2
R2c+R2g , (4.18) where κ0 is an epicyclic frequency,
λ = Λκ0, (4.19)
where λ is a damping rate and Λ is a constant value,
A≡ − 1
√∆2+ 16λ2(Ω0−Ωb)2 [(
dΦb
dR )
Rg
+ 2Ω0Φb(Rg) Rg(Ω0−Ωb)
]
, (4.20) and
A= 2aǫRg
(Rg2+ a2)2√
∆2+ 16λ2(Ω0−Ωb) + 4aǫRg3 (R2g+ a2)3+ R2Ω0Φb
g(Ω0−Ωb)
, (4.21)
∆ = κ20−4 (Ω0−Ωb)2, (4.22)
α= arctan
[ −2Λ κ0(Ω0 −Ωb)
]
, (4.23)
B =√
(E+F)2+ 2EF(cosα−1), (4.24)
E = Ω0 Ω0−Ωb
A, (4.25)
F = Φb(Rg)
2Rg(Ω0 −Ωb)2, (4.26)
β = arctan
( sinα cosα+F/E
)
. (4.27)
After (R, φ) is calculated, these are converted into the non-rotating galactic co-ordinate,
X = S1(Rcosφcosθ−Rsinφsinθ), (4.28) where S1 is a scale factor for the length,
Y = S1(Rcosφsinθ+Rsinφcosθ). (4.29) Thus the distance of each gas particle from the Sun is calculated with the distance between the Sun and the Galactic center (R0),
D=√
(R0+Y)2+X2, (4.30)
where the R0 is supposed to 8.0 kpc, and the galactic longitude is, l = arctan
( X R0+Y
)
. (4.31)
From equation 4.13 and 4.14, velocity in direction of R and φ are obtained,
VR= S2[−2(Ω0−Ωb)Asin (2φg+α)], (4.32)
Vφ= S2{(Ω0−Ωb)[Rg+Acos (2φg+α)−2Bcos (2φg+β)]}, (4.33)
where S2 is a scale factor for the velocity to convert to actual dimension of [km s−1].
These velocities are also converted to the Cartesian coordinates and rotate the bar major axis around θ(=θ′−90) degrees toward the Sun in the direction of the galactic longitude,
VX = (VRcosφ−Vφsinφ) cosθ′−(VRsinφ+Vφcosφ) sinθ′, (4.34)
VY = (VRcosφ−Vφsinφ) sinθ′+ (VRsinφ+Vφcosφ) cosθ′. (4.35) Finally, to obtain VLSR [km s−1] and µl [mas yr−1], VX and VY are convert to the galactic coordinate and subtracted the rotating speed of LSR (Θ0),
VLSR =VXsinl+VY cosl−Θ0sinl, (4.36)
µl = 0.2108
D (VXcosl−VY sinl−Θ0cosl), (4.37) where 0.2108 is a conversion factor to the dimension of [mas yr−1]. Here the galactic latitude are supposed to be b = 0.
Properties of this model are shown in figure 4.9 and 4.10. In these figures, several orbits of the damped oval orbit model (green colored plots) and the flat rotation model (red colored plots), the reduced chi-squares of each parameter and the rotation curve are plotted. Unlike the damped oval orbit model, the flat rotation model is shown include around corotation area at (X,Y) plots and (VLSR,µlcosb) plots. Here we assumed that v0 = 0.71 which is a dimensionless parameter. We also assumed that the S1 = 1.5 and the S2 = 310 which satisfy the l-v map of flat rotation.
To estimate the bar parameter of (Ωb, ǫ, θ), we estimate two kind of the reduced chi-square value (χ2ν). One is the χ2ν between above model and the observational data sets in table 4.6. Another is the χ2ν between above model and the terminal velocities of HI in table 4.7. Here we do not discuss about unknown parameter of Λ, and suppose Λ = 0.10 (c.f., Sakamoto et al. (1999)). χ2ν are calculated between ranges of 0.1 < Rg < 8.0 and −180◦ < φg < 180◦. When the amplitude of radial non-circular motion A in equation 4.21 becomes ∞ at the corotation radius, where
(Ω0−Ωb) = 0, the model considered here (which is based on linearized equations of small perturbations) becomes invalid. Therefore,we reject around this point by upper threshold of A = 0.37 and lower threshold of A = −0.37 to include inner Lindblad resonance and reject around corotation in this model calculation.
The best fit parameter is (Ωb, ǫ, θ) = (0.13, 0.10, 30◦) in the least-square fitting, and shown in figure 4.9 and top panel of figure 4.11. The Ωb of 0.13 is corresponding to 26.8 km s−1 kpc−1. Each value ofχ2ν are 0.8 and 7.0 at (VLSR,µlcosb) plot and (l, VLSR) plot, respectively. However, in this parameters, the radius of corotation (RCR) become 8.2 kpc, which is out of the Sun from the Galactic center. Generally, RCR
should be located in the inner part between the Sun and the Galactic center. This is too large when comparing with the previous studies.
The pattern speed of the Galactic bar are expected to be ∼50 - 60 km s−1 kpc−1 (Debattista et al. (2002); Dehnen (2000); Minchev et al. (2007)). If we used the general value of Ωb ∼ 55 km s−1 kpc−1 corresponding to the RCR ∼ 3 - 4.6 kpc, the regions where can not hold linearization are located in our sample of data within l ∼20 - 30◦. Thus, this model have limitations to discuss the motion with the typical galactic parameter of Ωbassociate withRCR. Thus, in this discussion with this model, we use the larger Ωb of 0.24 (which is corresponding to 49.7 km s−1 kpc−1) which is able to obtain smaller values of χ2ν. This value corresponds to aRCRof 4.4 kpc. This RCR value was consistent with the general values.
The results of comparison with the acceptable galactic parameters of (Ωb,ǫ, θ) = (0.24, 0.10, 50◦) are shown in figure 4.10 and bottom panel of figure 4.11. Values ofχ2ν are 1.2 and 4.0 at (VLSR,µlcosb) plot and (l,VLSR) plot, respectively. From the plots of χ2ν, we can see that the reasonable value of the galactic parameter ofθ ∼30◦- 50◦. This value is consistent with the value of 10◦- 50◦ obtained by completely other kind of observations at optical (e.g., Rattenbury et al. (2007a)).
From the value of Ωb = 0.24 or 0.13 around extreme in the χ2ν plot of Ωb for (VLSR, µlcosb) plot, this model suggest that peculiar motion between ILR and CR can explain the slower proper motions of VLBI data to the flat circular rotation. In figure 4.12 - 4.15, (VLSR,µlcosb) and (l,VLSR) diagrams and orbits on a (X,Y) plane
are shown every radius between each resonance. In these figures, a trend of slower proper motion to the flat circular rotation is remarkable tendency between ILR and CR, as shown in figure 4.14.
We show the summary of the values of χ2ν of each models in our discussions in table 4.8. The flat rotation model is able to reproduce thel-v map of the HIgas well.
However, this model is not suitable to reproduce the (VLSR, µlcosb) plot with slower proper motions deviating from circular motions. On the other hand, the non-flat rotation model is opposite situation. Only the damped oval orbit model was able to explain the l-v map and the (VLSR,µlcosb) plot well.
Table 4.8: Summary of χ2ν of each model.
χ2ν of (VLSR, µlcosb) χ2ν of (l, VLSR)
with VLBI data with HI data
Flat rotation 2.7 ◦ 10.4 ◦
Non-flat rotation 0.6 ⊚ 47.5 ×
Damped oval orbit 0.8 (best) / 1.2 (Ωb,accept) ⊚ 7.0 (best) / 4.0(Ωb,accept) ⊚ Therefore, introducing that the non-circular motion due to the Galactic bar is naturally explains the observed properties of maser proper motions as well as HI
terminal velocities. In addition, our model calculations based on the damped orbit model predict that the suitable parameter of the bar inclination is 30◦- 50◦, which is most affected by systemic velocity at each galactic longitude in the VLBI data, and consistent with other studies. Thus, it is rather natural to conclude that the our maser proper motions are also tracing the non-circular motions of the bar, and they provide another evidence of the bar, which is based on the 3-D motions of the gas for the first time.
In the future, we would like to get more samples toward wider range of the galactic longitude and get a trend of whole motions around the Galactic bar. If we compare with VLBI data and any models, we can not see distinct difference between each model, because each data accuracy is limited by amplitude of random motion in the
0 50 100 150 200
0 2 4 6 8 10 12
R [kpc]
-7 -6 -5 -4 -3 -2 -1 0
0 20 40 60 80 100 120 140 l=24°
-7 -6 -5 -4 -3 -2 -1 0
0 20 40 60 80 100 120 140
-7 -6 -5 -4 -3 -2 -1 0
0 20 40 60 80 100 120 140 l=28°
Circular orbits Damped oval orbits
-7 -6 -5 -4 -3 -2 -1 0
0 20 40 60 80 100 120 140 -7
-6 -5 -4 -3 -2 -1 0
0 20 40 60 80 100 120 140 -7
-6 -5 -4 -3 -2 -1 0
0 20 40 60 80 100 120 140 μlcosb [mas yr-1]
VLSR [km s-1] l=23°
Circular orbits Damped oval orbits
-4 -3 -2 -1 0 1 2 3 4
-4 -3 -2 -1 0 1 2 3 4
Y [kpc]
X [kpc]
Ωb=0.13, ε=0.10, θ=30°, Λ=0.1
-200 -100 0 100 200
-40 -30 -20 -10 0 10 20 30 40
l [deg]
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Ωb
ε=0.10, θ=30°
0 0.1 0.2 0.3 0.4 0.5
Xv2 of (VLSR,μlcosb)
ε Ωb=0.13, θ=30°
0 50 100 150
Xv2 of (l, Vterm)
ε Ωb=0.13,θ=30°
Ωb ε=0.10, θ=30°
θ [deg]
Ωb=26.8 km s-1 kpc-1 RILR=2.4 kpc RCR=8.2 kpc
VLSR [km s-1]Rotation Speed [km s-1] Pattern Speed [km s-1 kpc-1]
0 50 100 150 200
Circular orbits Damped oval orbits
Ωb=0.13,ε=0.10 θ [deg]
Ωb=0.13,ε=0.10
Xv2 of (VLSR,μlcosb)Xv2 of (l, Vterm) Xv2 of (VLSR,μlcosb)Xv2 of (l, Vterm)
l=29.9° l=30.7°
l=25°
Circular orbits Damped oval orbits
Circular orbits Damped oval orbits Circular orbits
Damped oval orbits
μlcosb [mas yr-1]
VLSR [km s-1]
μlcosb [mas yr-1]
VLSR [km s-1]
μlcosb [mas yr-1]
VLSR [km s-1]
μlcosb [mas yr-1]
VLSR [km s-1]
μlcosb [mas yr-1]
VLSR [km s-1]
Vc=RgΩ0 Ωb Ω0 Ω0-k0/2 Ω0+k0/2
5 10 15 20
0 0.2 0.4 0.6 0.8 1 0
50 100 150
0 0.1 0.2 0.3 0.4 0.5 5
10 15
0 50 100 150 200
0 5 10 15 20 25
0 5 10 15 20 25 30
0 1 2 3 4 5
85 6
R [kpc]
l=24°
l=28°
Circular orbits Damped oval orbits μlcosb [mas yr-1]
VLSR [km s-1] l=23°
Circular orbits Damped oval orbits
Y [kpc]
X [kpc]
Ωb=0.24, ε=0.10, θ=50°, Λ=0.1
l [deg]
Ωb ε=0.10, θ=50°
Xv2 of (VLSR,μlcosb)
ε Ωb=0.24, θ=50°
Xv2 of (l, Vterm)
ε Ωb=0.24,θ=50°
Ωb θ [deg]
Ωb=49.7 km s-1 kpc-1 RILR=1.3 kpc RCR=4.4 kpc ROLR=7.6 kpc
VLSR [km s-1]Rotation Speed [km s-1] Pattern Speed [km s-1 kpc-1]
0 50 100 150 200
Circular orbits Damped oval orbits
Ωb=0.24,ε=0.10 θ [deg]
Ωb=0.24,ε=0.10
Xv2 of (VLSR,μlcosb)Xv2 of (l, Vterm) Xv2 of (VLSR,μlcosb)Xv2 of (l, Vterm)
l=29.9° l=30.7°
l=25°
Circular orbits Damped oval orbits
Circular orbits Damped oval orbits Circular orbits
Damped oval orbits
μlcosb [mas yr-1]
VLSR [km s-1]
μlcosb [mas yr-1]μlcosb [mas yr-1]
VLSR [km s-1]
μlcosb [mas yr-1]μlcosb [mas yr-1]
VLSR [km s-1] -4
-3 -2 -1 0 1 2 3 4
-4 -3 -2 -1 0 1 2 3 4
-40 -30 -20 -10 0 10 20 30 40
0 50 100 150 200
0 2 4 6 8 10 12
Vc=RgΩ0 Ωb Ω0 Ω0-k0/2 Ω0+k0/2
-7 -6 -5 -4 -3 -2 -1 0
0 20 40 60 80 100 120 140 -7 -6 -5 -4 -3 -2 -1 0
-7 -6 -5 -4 -3 -2 -1 0
0 20 40 60 80 100 120 140
-7 -6 -5 -4 -3 -2 -1 0
0 20 40 60 80 100 120 140 -7 -6 -5 -4 -3 -2 -1 0
0 20 40 60 80 100 120 140 -7
-6 -5 -4 -3 -2 -1 0
ε=0.10, θ=50°
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5
0 0.1 0.2 0.3 0.4 0.5
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
VLSR [km s-1] VLSR [km s-1]
0 5 10 15
0 10 20 30 40 50 60 70
0 10 20 30 40
0 50 100 150
0 5 10 15
0 50 100 150 200
0 5 10 15 20 25
0 50 100 150 200
-200 -100 0 100 200
86
Damped oval orbit Flar circular motion
Damped oval orbit Flar circular motion
-200 -100 0 100 200
-40 -30 -20 -10 0 10 20 30 40
l [deg]
VLSR [km s-1]
l [deg]
VLSR [km s-1]
-200 -100 0 100 200
-40 -30 -20 -10 0 10 20 30 40
Terminal velocity
Terminal velocity
Ωb=0.13, ε=0.10, θ=30°, Λ=0.1
Ωb=0.24, ε=0.10, θ=50°, Λ=0.1
Figure 4.11: The longitude-velocity map of and HI, the flat circular rotation model and the damped oval orbit model with two parameter sets.